domain package:
* theorems are stored in the theory
* creates hierachical name space
* minor changes to some names and values (for consistency),
e.g. cases -> casedist, dists_eq -> dist_eqs,
[take_lemma] -> take_lemmas
* separator between mutual domain definitions changed from "," to "and"
* minor debugging of Domain_Library.mk_var_names
(* Title: HOLCF/Ssum2.ML
ID: $Id$
Author: Franz Regensburger
Copyright 1993 Technische Universitaet Muenchen
Lemmas for Ssum2.thy
*)
open Ssum2;
(* for compatibility with old HOLCF-Version *)
qed_goal "inst_ssum_po" thy "(op <<)=(%s1 s2.@z.\
\ (! u x. s1=Isinl u & s2=Isinl x --> z = u << x)\
\ &(! v y. s1=Isinr v & s2=Isinr y --> z = v << y)\
\ &(! u y. s1=Isinl u & s2=Isinr y --> z = (u = UU))\
\ &(! v x. s1=Isinr v & s2=Isinl x --> z = (v = UU)))"
(fn prems =>
[
(fold_goals_tac [less_ssum_def]),
(rtac refl 1)
]);
(* ------------------------------------------------------------------------ *)
(* access to less_ssum in class po *)
(* ------------------------------------------------------------------------ *)
qed_goal "less_ssum3a" thy "Isinl x << Isinl y = x << y"
(fn prems =>
[
(simp_tac (!simpset addsimps [less_ssum2a]) 1)
]);
qed_goal "less_ssum3b" thy "Isinr x << Isinr y = x << y"
(fn prems =>
[
(simp_tac (!simpset addsimps [less_ssum2b]) 1)
]);
qed_goal "less_ssum3c" thy "Isinl x << Isinr y = (x = UU)"
(fn prems =>
[
(simp_tac (!simpset addsimps [less_ssum2c]) 1)
]);
qed_goal "less_ssum3d" thy "Isinr x << Isinl y = (x = UU)"
(fn prems =>
[
(simp_tac (!simpset addsimps [less_ssum2d]) 1)
]);
(* ------------------------------------------------------------------------ *)
(* type ssum ++ is pointed *)
(* ------------------------------------------------------------------------ *)
qed_goal "minimal_ssum" thy "Isinl UU << s"
(fn prems =>
[
(res_inst_tac [("p","s")] IssumE2 1),
(hyp_subst_tac 1),
(rtac (less_ssum3a RS iffD2) 1),
(rtac minimal 1),
(hyp_subst_tac 1),
(stac strict_IsinlIsinr 1),
(rtac (less_ssum3b RS iffD2) 1),
(rtac minimal 1)
]);
bind_thm ("UU_ssum_def",minimal_ssum RS minimal2UU RS sym);
qed_goal "least_ssum" thy "? x::'a++'b.!y. x<<y"
(fn prems =>
[
(res_inst_tac [("x","Isinl UU")] exI 1),
(rtac (minimal_ssum RS allI) 1)
]);
(* ------------------------------------------------------------------------ *)
(* Isinl, Isinr are monotone *)
(* ------------------------------------------------------------------------ *)
qed_goalw "monofun_Isinl" thy [monofun] "monofun(Isinl)"
(fn prems =>
[
(strip_tac 1),
(etac (less_ssum3a RS iffD2) 1)
]);
qed_goalw "monofun_Isinr" thy [monofun] "monofun(Isinr)"
(fn prems =>
[
(strip_tac 1),
(etac (less_ssum3b RS iffD2) 1)
]);
(* ------------------------------------------------------------------------ *)
(* Iwhen is monotone in all arguments *)
(* ------------------------------------------------------------------------ *)
qed_goalw "monofun_Iwhen1" thy [monofun] "monofun(Iwhen)"
(fn prems =>
[
(strip_tac 1),
(rtac (less_fun RS iffD2) 1),
(strip_tac 1),
(rtac (less_fun RS iffD2) 1),
(strip_tac 1),
(res_inst_tac [("p","xb")] IssumE 1),
(hyp_subst_tac 1),
(asm_simp_tac Ssum0_ss 1),
(asm_simp_tac Ssum0_ss 1),
(etac monofun_cfun_fun 1),
(asm_simp_tac Ssum0_ss 1)
]);
qed_goalw "monofun_Iwhen2" thy [monofun] "monofun(Iwhen(f))"
(fn prems =>
[
(strip_tac 1),
(rtac (less_fun RS iffD2) 1),
(strip_tac 1),
(res_inst_tac [("p","xa")] IssumE 1),
(hyp_subst_tac 1),
(asm_simp_tac Ssum0_ss 1),
(asm_simp_tac Ssum0_ss 1),
(asm_simp_tac Ssum0_ss 1),
(etac monofun_cfun_fun 1)
]);
qed_goalw "monofun_Iwhen3" thy [monofun] "monofun(Iwhen(f)(g))"
(fn prems =>
[
(strip_tac 1),
(res_inst_tac [("p","x")] IssumE 1),
(hyp_subst_tac 1),
(asm_simp_tac Ssum0_ss 1),
(hyp_subst_tac 1),
(res_inst_tac [("p","y")] IssumE 1),
(hyp_subst_tac 1),
(asm_simp_tac Ssum0_ss 1),
(res_inst_tac [("P","xa=UU")] notE 1),
(atac 1),
(rtac UU_I 1),
(rtac (less_ssum3a RS iffD1) 1),
(atac 1),
(hyp_subst_tac 1),
(asm_simp_tac Ssum0_ss 1),
(rtac monofun_cfun_arg 1),
(etac (less_ssum3a RS iffD1) 1),
(hyp_subst_tac 1),
(res_inst_tac [("s","UU"),("t","xa")] subst 1),
(etac (less_ssum3c RS iffD1 RS sym) 1),
(asm_simp_tac Ssum0_ss 1),
(hyp_subst_tac 1),
(res_inst_tac [("p","y")] IssumE 1),
(hyp_subst_tac 1),
(res_inst_tac [("s","UU"),("t","ya")] subst 1),
(etac (less_ssum3d RS iffD1 RS sym) 1),
(asm_simp_tac Ssum0_ss 1),
(hyp_subst_tac 1),
(res_inst_tac [("s","UU"),("t","ya")] subst 1),
(etac (less_ssum3d RS iffD1 RS sym) 1),
(asm_simp_tac Ssum0_ss 1),
(hyp_subst_tac 1),
(asm_simp_tac Ssum0_ss 1),
(rtac monofun_cfun_arg 1),
(etac (less_ssum3b RS iffD1) 1)
]);
(* ------------------------------------------------------------------------ *)
(* some kind of exhaustion rules for chains in 'a ++ 'b *)
(* ------------------------------------------------------------------------ *)
qed_goal "ssum_lemma1" thy
"[|~(!i.? x. Y(i::nat)=Isinl(x))|] ==> (? i.! x. Y(i)~=Isinl(x))"
(fn prems =>
[
(cut_facts_tac prems 1),
(fast_tac HOL_cs 1)
]);
qed_goal "ssum_lemma2" thy
"[|(? i.!x.(Y::nat => 'a++'b)(i::nat)~=Isinl(x::'a))|] ==>\
\ (? i y. (Y::nat => 'a++'b)(i::nat)=Isinr(y::'b) & y~=UU)"
(fn prems =>
[
(cut_facts_tac prems 1),
(etac exE 1),
(res_inst_tac [("p","Y(i)")] IssumE 1),
(dtac spec 1),
(contr_tac 1),
(dtac spec 1),
(contr_tac 1),
(fast_tac HOL_cs 1)
]);
qed_goal "ssum_lemma3" thy
"[|is_chain(Y);(? i x. Y(i)=Isinr(x::'b) & (x::'b)~=UU)|] ==>\
\ (!i.? y. Y(i)=Isinr(y))"
(fn prems =>
[
(cut_facts_tac prems 1),
(etac exE 1),
(etac exE 1),
(rtac allI 1),
(res_inst_tac [("p","Y(ia)")] IssumE 1),
(rtac exI 1),
(rtac trans 1),
(rtac strict_IsinlIsinr 2),
(atac 1),
(etac exI 2),
(etac conjE 1),
(res_inst_tac [("m","i"),("n","ia")] nat_less_cases 1),
(hyp_subst_tac 2),
(etac exI 2),
(eres_inst_tac [("P","x=UU")] notE 1),
(rtac (less_ssum3d RS iffD1) 1),
(eres_inst_tac [("s","Y(i)"),("t","Isinr(x)::'a++'b")] subst 1),
(eres_inst_tac [("s","Y(ia)"),("t","Isinl(xa)::'a++'b")] subst 1),
(etac (chain_mono RS mp) 1),
(atac 1),
(eres_inst_tac [("P","xa=UU")] notE 1),
(rtac (less_ssum3c RS iffD1) 1),
(eres_inst_tac [("s","Y(i)"),("t","Isinr(x)::'a++'b")] subst 1),
(eres_inst_tac [("s","Y(ia)"),("t","Isinl(xa)::'a++'b")] subst 1),
(etac (chain_mono RS mp) 1),
(atac 1)
]);
qed_goal "ssum_lemma4" thy
"is_chain(Y) ==> (!i.? x. Y(i)=Isinl(x))|(!i.? y. Y(i)=Isinr(y))"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac case_split_thm 1),
(etac disjI1 1),
(rtac disjI2 1),
(etac ssum_lemma3 1),
(rtac ssum_lemma2 1),
(etac ssum_lemma1 1)
]);
(* ------------------------------------------------------------------------ *)
(* restricted surjectivity of Isinl *)
(* ------------------------------------------------------------------------ *)
qed_goal "ssum_lemma5" thy
"z=Isinl(x)==> Isinl((Iwhen (LAM x. x) (LAM y. UU))(z)) = z"
(fn prems =>
[
(cut_facts_tac prems 1),
(hyp_subst_tac 1),
(case_tac "x=UU" 1),
(asm_simp_tac Ssum0_ss 1),
(asm_simp_tac Ssum0_ss 1)
]);
(* ------------------------------------------------------------------------ *)
(* restricted surjectivity of Isinr *)
(* ------------------------------------------------------------------------ *)
qed_goal "ssum_lemma6" thy
"z=Isinr(x)==> Isinr((Iwhen (LAM y. UU) (LAM x. x))(z)) = z"
(fn prems =>
[
(cut_facts_tac prems 1),
(hyp_subst_tac 1),
(case_tac "x=UU" 1),
(asm_simp_tac Ssum0_ss 1),
(asm_simp_tac Ssum0_ss 1)
]);
(* ------------------------------------------------------------------------ *)
(* technical lemmas *)
(* ------------------------------------------------------------------------ *)
qed_goal "ssum_lemma7" thy
"[|Isinl(x) << z; x~=UU|] ==> ? y. z=Isinl(y) & y~=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(res_inst_tac [("p","z")] IssumE 1),
(hyp_subst_tac 1),
(etac notE 1),
(rtac antisym_less 1),
(etac (less_ssum3a RS iffD1) 1),
(rtac minimal 1),
(fast_tac HOL_cs 1),
(hyp_subst_tac 1),
(rtac notE 1),
(etac (less_ssum3c RS iffD1) 2),
(atac 1)
]);
qed_goal "ssum_lemma8" thy
"[|Isinr(x) << z; x~=UU|] ==> ? y. z=Isinr(y) & y~=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(res_inst_tac [("p","z")] IssumE 1),
(hyp_subst_tac 1),
(etac notE 1),
(etac (less_ssum3d RS iffD1) 1),
(hyp_subst_tac 1),
(rtac notE 1),
(etac (less_ssum3d RS iffD1) 2),
(atac 1),
(fast_tac HOL_cs 1)
]);
(* ------------------------------------------------------------------------ *)
(* the type 'a ++ 'b is a cpo in three steps *)
(* ------------------------------------------------------------------------ *)
qed_goal "lub_ssum1a" thy
"[|is_chain(Y);(!i.? x. Y(i)=Isinl(x))|] ==>\
\ range(Y) <<|\
\ Isinl(lub(range(%i.(Iwhen (LAM x. x) (LAM y. UU))(Y i))))"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac is_lubI 1),
(rtac conjI 1),
(rtac ub_rangeI 1),
(rtac allI 1),
(etac allE 1),
(etac exE 1),
(res_inst_tac [("t","Y(i)")] (ssum_lemma5 RS subst) 1),
(atac 1),
(rtac (monofun_Isinl RS monofunE RS spec RS spec RS mp) 1),
(rtac is_ub_thelub 1),
(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
(strip_tac 1),
(res_inst_tac [("p","u")] IssumE2 1),
(res_inst_tac [("t","u")] (ssum_lemma5 RS subst) 1),
(atac 1),
(rtac (monofun_Isinl RS monofunE RS spec RS spec RS mp) 1),
(rtac is_lub_thelub 1),
(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
(etac (monofun_Iwhen3 RS ub2ub_monofun) 1),
(hyp_subst_tac 1),
(rtac (less_ssum3c RS iffD2) 1),
(rtac chain_UU_I_inverse 1),
(rtac allI 1),
(res_inst_tac [("p","Y(i)")] IssumE 1),
(asm_simp_tac Ssum0_ss 1),
(asm_simp_tac Ssum0_ss 2),
(etac notE 1),
(rtac (less_ssum3c RS iffD1) 1),
(res_inst_tac [("t","Isinl(x)")] subst 1),
(atac 1),
(etac (ub_rangeE RS spec) 1)
]);
qed_goal "lub_ssum1b" thy
"[|is_chain(Y);(!i.? x. Y(i)=Isinr(x))|] ==>\
\ range(Y) <<|\
\ Isinr(lub(range(%i.(Iwhen (LAM y. UU) (LAM x. x))(Y i))))"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac is_lubI 1),
(rtac conjI 1),
(rtac ub_rangeI 1),
(rtac allI 1),
(etac allE 1),
(etac exE 1),
(res_inst_tac [("t","Y(i)")] (ssum_lemma6 RS subst) 1),
(atac 1),
(rtac (monofun_Isinr RS monofunE RS spec RS spec RS mp) 1),
(rtac is_ub_thelub 1),
(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
(strip_tac 1),
(res_inst_tac [("p","u")] IssumE2 1),
(hyp_subst_tac 1),
(rtac (less_ssum3d RS iffD2) 1),
(rtac chain_UU_I_inverse 1),
(rtac allI 1),
(res_inst_tac [("p","Y(i)")] IssumE 1),
(asm_simp_tac Ssum0_ss 1),
(asm_simp_tac Ssum0_ss 1),
(etac notE 1),
(rtac (less_ssum3d RS iffD1) 1),
(res_inst_tac [("t","Isinr(y)")] subst 1),
(atac 1),
(etac (ub_rangeE RS spec) 1),
(res_inst_tac [("t","u")] (ssum_lemma6 RS subst) 1),
(atac 1),
(rtac (monofun_Isinr RS monofunE RS spec RS spec RS mp) 1),
(rtac is_lub_thelub 1),
(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
(etac (monofun_Iwhen3 RS ub2ub_monofun) 1)
]);
bind_thm ("thelub_ssum1a", lub_ssum1a RS thelubI);
(*
[| is_chain ?Y1; ! i. ? x. ?Y1 i = Isinl x |] ==>
lub (range ?Y1) = Isinl
(lub (range (%i. Iwhen (LAM x. x) (LAM y. UU) (?Y1 i))))
*)
bind_thm ("thelub_ssum1b", lub_ssum1b RS thelubI);
(*
[| is_chain ?Y1; ! i. ? x. ?Y1 i = Isinr x |] ==>
lub (range ?Y1) = Isinr
(lub (range (%i. Iwhen (LAM y. UU) (LAM x. x) (?Y1 i))))
*)
qed_goal "cpo_ssum" thy
"is_chain(Y::nat=>'a ++'b) ==> ? x. range(Y) <<|x"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac (ssum_lemma4 RS disjE) 1),
(atac 1),
(rtac exI 1),
(etac lub_ssum1a 1),
(atac 1),
(rtac exI 1),
(etac lub_ssum1b 1),
(atac 1)
]);